1. Henrico County Public Schools New Mathematics Teachers General Math Session How to make your horse thirsty and other stuff you might want to know
August 24, :30 p.m. – 4:30 p.m. Hermitage High School

2. Welcome! Introductions Name, School, Teaching Experience
What brought you to HCPS?
What made you decide to teach math?

3. Objectives To familiarize you with curriculum expectations
To introduce you to the materials, places, people and processes that can help you do your best
To teach you how to get what you want

4. Today’s Schedule It’s pretty simple… General Information Break
Collaborative Teaching/Differentiation
Breakout Sessions
Middle School
High School

6. Planning
VDOE Website:
Curriculum Framework – have a copy on your desk!
Blueprints
Keep in mind, SOLs are the MINIMUM standard.

7. If you teach to the SOL test…

8. Standards of Learning
The Standards of Learning provide a good framework. However, teaching is more than just learning an SOL!

9. 2009 Mathematics Standards of Learning
Rigor has been increased
Repetition has been decreased
Retention and application of content from previous years required
Vertical alignment has been improved

10. Vertical Articulation Documents
Click here for documents
Vertical Articulation Documents

11. Vertical Articulation of Content
Why is it important knowledge to have?
Consistency
Connections
Relevance
The Mathematics Crosswalk Between the 2009 and 2001 Standards (PDF) provides detail on additions, deletions and changes included in the 2009 Mathematics Standards of Learning.
All these lead to deeper understanding and long-term retention of content

12. Pay attention to details!

13. Course Resources and Pacing Guide
All digital curriculum and pacing guides are available on the HCPS math website:

14. HCPS Teacher Resource Page
A lot of links to important documents!
ExamView test banks
Online textbooks
Carnegie Learning files
Promethean & ActivEngage information
NTA powerpoint
Graphing calculator instructions

15. Speaking of calculators…
Make sure that the calculator is a TOOL used for instruction!
It has become a crutch for many students and teachers.
Be accountable for them and have a system for storing and collecting them.

16. Observations/Evaluations
“Snapshot” observations
Teacher requested observations
Formal Evaluation Process

17. People who can help you! YOU Specialist Assistant Colleagues Principal
Mentor or Buddy
Dept
Chair
Assistant
Principal
Specialist
Colleagues
ITRT
Coaches

18. You can lead a horse to water, but you can’t make him drink
In the past, an acceptable philosophy for educators was:
You can lead a horse to water, but you can’t make him drink

19. The present philosophy for educators goes something like this:
If you lead a horse to water and he doesn’t want to drink, it’s your job to make him thirsty.

20. Instruction Make lessons active Have fun learning!
Limit lecture – the least effective teaching method
Model the skills that you want your students to exhibit
Develop concepts rather than answers
Vary your teaching strategies
Challenge the students - rigor
Have fun learning!
Encourage students to participate
Set the stage for student success

21. A critical point
…a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a task for, and some means of, independent thinking.”
Polya, 1973/1945

22. Problem Based Learning
The Die Hard with a Vengeance version of teaching Polya's four-step problem solving process

23. Getting Started & Surviving the first month!
NCTM – tips for teachers -
Develop a support system
In your department
In your school
In your county
*In your family*

24. Manage Your Time Wisely
Planning
Develop a schedule
The first year in any job is the hardest
Plan effective lessons
Make objectives clear to the students
Routine and structure are good, but don’t fall into a rut. Try to vary activities from time to time.
Be prepared for the unexpected. It will happen. Daily.

25. Don’t send mixed messages. Be consistent!
Classroom Management
Positive first impressions
Greet and welcome your students as they enter the room
Have a plan for the class
Share your enthusiasm
Help students to be successful
Use positive reinforcement to motivate students - give out awards for both good academics and for good effort
Have structure and procedures
With these in place discipline follows
You don’t want students creating classroom rules on the fly
Fewer rules are better
Be fair (fair does not mean equal)
Don’t send mixed messages. Be consistent!

26. Assessing Instruction
Assessment
More than tests and quizzes
Assessment for Learning – ActivEngage!
Spell out what topics will be on the test. This will especially help those with poor study skills.
ExamView banks
Grades
Interims, Quarters, Semesters
eClass grading program
Technology
Graphing and Scientific Calculators
Interims weeks - benchmark
Quarters - 9 weeks and used in determining semester grades
Semesters weeks and used in determining final grades
Why give grades less than 50%?
Total Points vs Categories

27. Changing Instruction Dan Meyer – Math class needs a makeover
Today's math curriculum is teaching students to expect -- and excel at -- paint-by-numbers classwork, robbing kids of a skill more important than solving problems: formulating them. At TEDxNYED, Dan Meyer shows classroom-tested math exercises that prompt students to stop and think.

28. Questions? Please contact me about anything! Skip Tyler

29. Break

30. Differentiate in the classroom
Differentiate in the classroom. Realize that students have different skills sets.

31. Collaborative Teaching/Differentiated Learning
Ms. Ashley Reyher and Mr. Kevin Hoy
Collaborative Math Classrooms:  Co-teaching Tips and Strategies
Collaborative Math flipchart

32. Breakout sessions High school stays here
Middle school goes to room 160

33. Examine the EOC Vertical Articulation
Identify the similarities and differences between the grade levels
What are the key verbs?
Was there anything that surprised you?
Breaking Down the Standards
List the 5 most important concepts you see in your standards
Can you draw a representation of the topics?
mathematics/index.shtml

34. Course 1 Standards
Grade 6

35. Course 2 Standards
Grade 7

36. Course 3 Standards
Grade 8

37. Algebra 1 Standards

38. Geometry Standards

39. Algebra 2 Standards

40. Instruction: Why focus on tasks?
Classroom instruction is generally organized and orchestrated around mathematical tasks
The tasks with which students engage determines what they learn about mathematics and how they learn it
“There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics” Lappan & Briars, 1995
Rich Mathematical Tasks: The Cornerstone of Rigorous Instruction

41. Instruction: Two tasks.
Martha’s Carpeting Task
Martha was re-carpeting her bedroom which was 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase?
Fencing Task
Ms. Brown’s class will raise rabbits for
their spring science fair. They have 24
feet of fencing with which to build a
rectangular rabbit pen in which to keep
the rabbits.
If Ms. Brown's students want their rabbits to have as much room as possible, how long would each of the sides of the pen be?
How long would each of the sides of the pen be if they had only 16 feet of fencing?
How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it.
Rich Mathematical Tasks:
Fencing Task
Build pens with physical materials (linear and area pieces)
Draw pens on grid paper (grid paper)
Make a table of the dimensions of possible pens
Make a graph that shows the relationship between one linear dimension and the area (graph paper or graphing calculator)
Set up an algebraic equation and solve

42. Instruction: Tasks - Comparison
Similarities
Both require prior knowledge of area
Area problems
Differences
Way in which the area formula is used
The need to generalize
The amount of thinking and reasoning required
The number of ways the problem can be solved
The range of ways to enter the problem
Way in which area formula is used -Martha’s Carpeting can be solved by knowing and using the area formula but this formula alone is not sufficient to solve the Fencing Task
- Martha’s carpeting does not lead to a generalization but the Fencing Task does
The amount of thinking and reasoning required - Martha’s Carpeting requires limit thinking and reasoning while the Fencing Task can’t be solved without it
Fencing task might be considered a more equitable task -- for several reasons…
Build pens with physical materials (linear and area pieces)
Draw pens on grid paper (grid paper)
Make a table of the dimensions of possible pens
Make a graph that shows the relationship between one linear dimension and the area (graph paper or graphing calculator)
Set up an algebraic equation and solve

43. Instruction: Tasks
Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.
Stein, Smith, Henningsen, & Silver, 2000
The level and kind of thinking in which students engage determines what they will learn.
Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997
If we want students to develop the capacity to think, reason, and problem solve then we need to start with high-level, cognitively complex tasks.
Stein & Lane, 1996

44. Task Characteristics High cognitive demand
Significant content (i.e., they have the potential to leave behind important residue)
Require justification or explanation
Make connections between two or more representations
Open-ended
Allow entry to students with a range of skills and abilities
Multiple ways to show competence
These last two characteristics can help make a task EQUITABLE -- That is, level the playing field so that students with a wide range of abilities can ENTER a TASK.

45. Tying it All Together
Improved vertical alignment of content with increased cognitive demand.
Key conceptual models can be extended across grade levels.
Refer to the Curriculum Framework.
Pay attention to the changes in the verbs. 45

46. Biggest changes!
Algebra 1, AFDA, and Algebra 2 have new information related to statistics.
Standard Deviation
Z-scores
Normal Distribution
Many students did not have this taught last year. Extra time and training will be necessary.

47. Four Scenarios!!!
Read the scenario presented Think through and jot down answers Share ideas/Ask Questions

48. Scenario #1
You were hired to teach Algebra 1. When you arrive at your new school your department leader informs you that you will need to teach one section of _____________ (something you are not comfortable teaching) in addition to the Algebra 1 classes.
What can you do?
What are some possible solutions?
Who can assist you with this situation?

49. Scenario #2
As a new mathematics teacher, you are assigned a mentor/buddy and are somewhat nervous about your teaching assignment. You are surprised to learn that your mentor, though very nice, does not teach your content. You also learn that the only other teacher who teaches your content is an old grouch who does nothing but complain.
What can you do?
What are some possible solutions?
Who can assist you with this situation?

50. Scenario #3
You will be teaching Geometry in a relatively new school that has a rapidly growing population. Unfortunately, the previous teacher took all of the supplies from your classroom. You mention this to teacher next door, whose room is well stocked with textbooks and calculators. She says that you can borrow her stuff when you need it, as long as you give her a week advance notice and return the materials within 24 hours after you are finished with it.
What can you do?
What are some possible solutions?
Who can assist you with this situation?

51. Scenario #4
You are a veteran teacher with 20 years of teaching experience. You take great pride in your ability to teach and were highly valued as a master teacher in your previous school division. However, you taught in a rural classroom and have very little experience with computers. Eclass and Winschool overwhelm you. All of the young teachers appear to be catching on quickly, and you are getting further and further behind.
What can you do?
What are some possible solutions?
Who can assist you with this situation?

52. NCTM Process Standards
Communication
Connections
Problem Solving
Reasoning and Proof
Representation

53. NCTM Process Standards
Communication
Use the language of mathematics to express mathematical ideas precisely.  
Writing about mathematics.
Discussing mathematical ideas.
You have eleven fruits in your basket, some are one kind of fruit, and the rest are another kind. How many of each could you have?

54. NCTM Process Standards
Connections
Between different mathematical areas
Between mathematics and science
Between mathematics and other subject areas (such as history, literature, and art)
Between mathematics and the real world
Example
Mr. Goodlock drives to and from Hermitage almost every day. Along the way the posted speed limits range from 30 mph to 65 mph. Mr. Goodlock has logged his daily commute.

55. NCTM Process Standards
Odometer Reading
Posted
MPH 60
2.7 55
7.8 45
11.8 35
18.2 65
24.3 30
27.3
Parking Lot
Connections
These are the posted speed limits and the odometer reading at the beginning of each drive segment Mr. Goodlock encounters:
Calculate the amount of time Mr. Goodlock spends in each speed zone. Make a graph showing your results. Mr. Goodlock usually drives at the posted speed limit. If we assume a trip with no traffic and we ignore time spent at stop signs and traffic lights -- what is the total driving time for Mr. Goodlock's trip?

56. WCYDWT Video (What Can You Do With This)
The best motivator of all is connecting math to the real world.

57. NCTM Process Standards
Problem Solving
Build new mathematical knowledge through problem solving
Solve problems that arise in mathematics and in other contexts 
Apply and adapt a variety of appropriate strategies to solve problems 
Monitor and reflect on the process of mathematical problem solving 
Translation: include one and two-step story problems typically found in textbooks.
A school auditorium can seat 648 people in 18 equal rows. How many seats are there in each row?
Process: requires solution processes other than computational procedures.
Application: computation is generally the solution process used to solve application problems.
How many soda cans would it take to fill the school gym?
New Door Problem - Rearrange the 7 letters in New Door to form one word
Process - At an air show, 8 skydivers were released from a plane. Each skydiver was connected to each of the other skydivers with a separate piece of ribbon. How many pieces of ribbon were used in the skydiving act?

58. NCTM Process Standards
Reasoning & Proof
Recognize reasoning and proof as fundamental aspects of mathematics 
Make and investigate mathematical conjectures 
Develop and evaluate mathematical arguments and proofs 
Select and use various types of reasoning and methods of proof 
Reasoning
Look at the set shown below.
{15, 23, 39, 42}
Which number is prime?
Reasoning
Look at the set shown below.
{2a, 3a, 4a, 5a}
If a is a prime number, how many members of the set are also prime?
From 2004 Grade 8 SOL test

59. NCTM Process Standards
Mathematical Representations
Create and use representations to organize, record, and communicate mathematical ideas 
Select, apply, and translate among mathematical representations to solve problems 
Use representations to model and interpret physical, social, and mathematical phenomena 
Sam went to a store and spent half of his money. Then he gave one-fifth of what he had left to his sister. Of the amount he had left, he lost half of it. When Sam got home, he had $ How much money did he have before entering the store?
Sam went to a store and spent half of his money. Then he gave one-fifth of what he had left to his sister. Of the amount he had left, he lost half of it. When Sam got home, he had $ How much money did he have before entering the store?

60. Bloom’s Taxonomy

61. Bloom’s Taxonomy
What is the mathematics assessed in the item on the right?
Which cognitive level does the question address?
76, 79, 75, 77, For the data listed, the value 76.2 represents the
A. Median
B. Mode
C. Range
D. Mean
Grade SOL test, Analysis
Ans. D

62. Bloom’s Taxonomy
The difference in cost between a large
bag of chips and a small bag of chips
was $.90. Alicia bought 5 large bags and
3 small bags of chips for her party and
spent $ What was the cost of a
small bag of chips?

F $5.74

G $2.49

H $2.15
What is the mathematics assessed in the item on the right?
Which cognitive level does the question address?
2006 SOL Algebra 1 test, systems
Ans. J

63. Effective Questioning
Is 15 a prime number?
Students can answer with a simple Yes or No.
What does the student’s response inform the teacher about the pupil’s knowledge about prime numbers?
Why is 7 an example of a prime number?
Not a one-word answer.
This requires a student to recall prior knowledge to explain and justify their reasoning.
It provides an opportunity to make an assessment without necessarily asking supplementary questions. The question, Is 15 a prime number? requires follow-up questions to get a full response on which to make an assessment.
The question Why is 7 an example of a prime number? is an example of the general question Why is x an example of y? This is one type of question that is effective in providing assessment opportunities.

64. Examine the 5-8 Vertical Articulation
Identify the similarities and differences between the grade levels
What are the key verbs?
Was there anything that surprised you?
Breaking Down the 6-8 Standards
List the 5 most important concepts you see in Grades 6 and 7
Can you draw a representation of the topics?

65. Number and Number Sense & Computation and Estimation
Grade 6

66. Number and Number Sense & Computation and Estimation
Grade 7

67. Number and Number Sense & Computation and Estimation
Grade 8

68. Measurement and Geometry
Grade 6

69. Measurement and Geometry
Grade 7

70. Measurement and Geometry
Grade 8

71. Probability, Statistics, Patterns, Functions, and Algebra
Grade 6

72. Probability, Statistics, Patterns, Functions, and Algebra
Grade 7

73. Probability, Statistics, Patterns, Functions, and Algebra
Grade 8

74. Parting words of wisdom…

75. Try to anticipate how a student might misuse equipment

76. Encourage your students to pay attention and make detailed observations.

77. Make sure there is substance to your lesson.

78. Set realistic expectations for your students